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Fixed point theorems for nonexpansive mappings in modular spaces

Poom Kumam (2004)

Archivum Mathematicum

In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if ρ is a convex, ρ -complete modular space satisfying the Fatou property and ρ r -uniformly convex for all r > 0 , C a convex, ρ -closed, ρ -bounded subset of X ρ , T : C C a ρ -nonexpansive mapping, then T has a fixed point.

Fixed points of asymptotically regular mappings in spaces with uniformly normal structure

Jarosław Górnicki (1991)

Commentationes Mathematicae Universitatis Carolinae

It is proved that: for every Banach space X which has uniformly normal structure there exists a k > 1 with the property: if A is a nonempty bounded closed convex subset of X and T : A A is an asymptotically regular mapping such that lim inf n | | | T n | | | < k , where | | | T | | | is the Lipschitz constant (norm) of T , then T has a fixed point in A .

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